The Stacks project

42.27 The key formula

Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Assume $X$ is integral and $\dim _\delta (X) = n$. Let $\mathcal{L}$ and $\mathcal{N}$ be invertible sheaves on $X$. Let $s$ be a nonzero meromorphic section of $\mathcal{L}$ and let $t$ be a nonzero meromorphic section of $\mathcal{N}$. Let $Z_ i \subset X$, $i \in I$ be a locally finite set of irreducible closed subsets of codimension $1$ with the following property: If $Z \not\in \{ Z_ i\} $ with generic point $\xi $, then $s$ is a generator for $\mathcal{L}_\xi $ and $t$ is a generator for $\mathcal{N}_\xi $. Such a set exists by Divisors, Lemma 31.27.2. Then

\[ \text{div}_\mathcal {L}(s) = \sum \text{ord}_{Z_ i, \mathcal{L}}(s) [Z_ i] \]

and similarly

\[ \text{div}_\mathcal {N}(t) = \sum \text{ord}_{Z_ i, \mathcal{N}}(t) [Z_ i] \]

Unwinding the definitions more, we pick for each $i$ generators $s_ i \in \mathcal{L}_{\xi _ i}$ and $t_ i \in \mathcal{N}_{\xi _ i}$ where $\xi _ i$ is the generic point of $Z_ i$. Then we can write

\[ s = f_ i s_ i \quad \text{and}\quad t = g_ i t_ i \]

Set $B_ i = \mathcal{O}_{X, \xi _ i}$. Then by definition

\[ \text{ord}_{Z_ i, \mathcal{L}}(s) = \text{ord}_{B_ i}(f_ i) \quad \text{and}\quad \text{ord}_{Z_ i, \mathcal{N}}(t) = \text{ord}_{B_ i}(g_ i) \]

Since $t_ i$ is a generator of $\mathcal{N}_{\xi _ i}$ we see that its image in the fibre $\mathcal{N}_{\xi _ i} \otimes \kappa (\xi _ i)$ is a nonzero meromorphic section of $\mathcal{N}|_{Z_ i}$. We will denote this image $t_ i|_{Z_ i}$. From our definitions it follows that

\[ c_1(\mathcal{N}) \cap \text{div}_\mathcal {L}(s) = \sum \text{ord}_{B_ i}(f_ i) (Z_ i \to X)_*\text{div}_{\mathcal{N}|_{Z_ i}}(t_ i|_{Z_ i}) \]

and similarly

\[ c_1(\mathcal{L}) \cap \text{div}_\mathcal {N}(t) = \sum \text{ord}_{B_ i}(g_ i) (Z_ i \to X)_*\text{div}_{\mathcal{L}|_{Z_ i}}(s_ i|_{Z_ i}) \]

in $\mathop{\mathrm{CH}}\nolimits _{n - 2}(X)$. We are going to find a rational equivalence between these two cycles. To do this we consider the tame symbol

\[ \partial _{B_ i}(f_ i, g_ i) \in \kappa (\xi _ i)^* \]

see Section 42.5.

Lemma 42.27.1 (Key formula). In the situation above the cycle

\[ \sum (Z_ i \to X)_*\left( \text{ord}_{B_ i}(f_ i) \text{div}_{\mathcal{N}|_{Z_ i}}(t_ i|_{Z_ i}) - \text{ord}_{B_ i}(g_ i) \text{div}_{\mathcal{L}|_{Z_ i}}(s_ i|_{Z_ i}) \right) \]

is equal to the cycle

\[ \sum (Z_ i \to X)_*\text{div}(\partial _{B_ i}(f_ i, g_ i)) \]

Proof. First, let us examine what happens if we replace $s_ i$ by $us_ i$ for some unit $u$ in $B_ i$. Then $f_ i$ gets replaced by $u^{-1} f_ i$. Thus the first part of the first expression of the lemma is unchanged and in the second part we add

\[ -\text{ord}_{B_ i}(g_ i)\text{div}(u|_{Z_ i}) \]

(where $u|_{Z_ i}$ is the image of $u$ in the residue field) by Divisors, Lemma 31.27.3 and in the second expression we add

\[ \text{div}(\partial _{B_ i}(u^{-1}, g_ i)) \]

by bi-linearity of the tame symbol. These terms agree by property (6) of the tame symbol.

Let $Z \subset X$ be an irreducible closed with $\dim _\delta (Z) = n - 2$. To show that the coefficients of $Z$ of the two cycles of the lemma is the same, we may do a replacement $s_ i \mapsto us_ i$ as in the previous paragraph. In exactly the same way one shows that we may do a replacement $t_ i \mapsto vt_ i$ for some unit $v$ of $B_ i$.

Since we are proving the equality of cycles we may argue one coefficient at a time. Thus we choose an irreducible closed $Z \subset X$ with $\dim _\delta (Z) = n - 2$ and compare coefficients. Let $\xi \in Z$ be the generic point and set $A = \mathcal{O}_{X, \xi }$. This is a Noetherian local domain of dimension $2$. Choose generators $\sigma $ and $\tau $ for $\mathcal{L}_\xi $ and $\mathcal{N}_\xi $. After shrinking $X$, we may and do assume $\sigma $ and $\tau $ define trivializations of the invertible sheaves $\mathcal{L}$ and $\mathcal{N}$ over all of $X$. Because $Z_ i$ is locally finite after shrinking $X$ we may assume $Z \subset Z_ i$ for all $i \in I$ and that $I$ is finite. Then $\xi _ i$ corresponds to a prime $\mathfrak q_ i \subset A$ of height $1$. We may write $s_ i = a_ i \sigma $ and $t_ i = b_ i \tau $ for some $a_ i$ and $b_ i$ units in $A_{\mathfrak q_ i}$. By the remarks above, it suffices to prove the lemma when $a_ i = b_ i = 1$ for all $i$.

Assume $a_ i = b_ i = 1$ for all $i$. Then the first expression of the lemma is zero, because we choose $\sigma $ and $\tau $ to be trivializing sections. Write $s = f\sigma $ and $t = g \tau $ with $f$ and $g$ in the fraction field of $A$. By the previous paragraph we have reduced to the case $f_ i = f$ and $g_ i = g$ for all $i$. Moreover, for a height $1$ prime $\mathfrak q$ of $A$ which is not in $\{ \mathfrak q_ i\} $ we have that both $f$ and $g$ are units in $A_\mathfrak q$ (by our choice of the family $\{ Z_ i\} $ in the discussion preceding the lemma). Thus the coefficient of $Z$ in the second expression of the lemma is

\[ \sum \nolimits _ i \text{ord}_{A/\mathfrak q_ i}(\partial _{B_ i}(f, g)) \]

which is zero by the key Lemma 42.6.3. $\square$

Remark 42.27.2. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $k \in \mathbf{Z}$. We claim that there is a complex

\[ \bigoplus \nolimits _{\delta (x) = k + 2}' K_2^ M(\kappa (x)) \xrightarrow {\partial } \bigoplus \nolimits _{\delta (x) = k + 1}' K_1^ M(\kappa (x)) \xrightarrow {\partial } \bigoplus \nolimits _{\delta (x) = k}' K_0^ M(\kappa (x)) \]

Here we use notation and conventions introduced in Remark 42.19.2 and in addition

  1. $K_2^ M(\kappa (x))$ is the degree $2$ part of the Milnor K-theory of the residue field $\kappa (x)$ of the point $x \in X$ (see Remark 42.6.4) which is the quotient of $\kappa (x)^* \otimes _\mathbf {Z} \kappa (x)^*$ by the subgroup generated by elements of the form $\lambda \otimes (1 - \lambda )$ for $\lambda \in \kappa (x) \setminus \{ 0, 1\} $, and

  2. the first differential $\partial $ is defined as follows: given an element $\xi = \sum _ x \alpha _ x$ in the first term we set

    \[ \partial (\xi ) = \sum \nolimits _{x \leadsto x',\ \delta (x') = k + 1} \partial _{\mathcal{O}_{W_ x, x'}}(\alpha _ x) \]

    where $\partial _{\mathcal{O}_{W_ x, x'}} : K_2^ M(\kappa (x)) \to K_1^ M(\kappa (x))$ is the tame symbol constructed in Section 42.5.

We claim that we get a complex, i.e., that $\partial \circ \partial = 0$. To see this it suffices to take an element $\xi $ as above and a point $x'' \in X$ with $\delta (x'') = k$ and check that the coefficient of $x''$ in the element $\partial (\partial (\xi ))$ is zero. Because $\xi = \sum \alpha _ x$ is a locally finite sum, we may in fact assume by additivity that $\xi = \alpha _ x$ for some $x \in X$ with $\delta (x) = k + 2$ and $\alpha _ x \in K_2^ M(\kappa (x))$. By linearity again we may assume that $\alpha _ x = f \otimes g$ for some $f, g \in \kappa (x)^*$. Denote $W \subset X$ the integral closed subscheme with generic point $x$. If $x'' \not\in W$, then it is immediately clear that the coefficient of $x$ in $\partial (\partial (\xi ))$ is zero. If $x'' \in W$, then we see that the coefficient of $x''$ in $\partial (\partial (x))$ is equal to

\[ \sum \nolimits _{x \leadsto x' \leadsto x'',\ \delta (x') = k + 1} \text{ord}_{\mathcal{O}_{\overline{\{ x'\} }, x''}}( \partial _{\mathcal{O}_{W, x'}}(f, g)) \]

The key algebraic Lemma 42.6.3 says exactly that this is zero.

Remark 42.27.3. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Let $k \in \mathbf{Z}$. The complex in Remark 42.27.2 and the presentation of $\mathop{\mathrm{CH}}\nolimits _ k(X)$ in Remark 42.19.2 suggests that we can define a first higher Chow group

\[ \mathop{\mathrm{CH}}\nolimits ^ M_ k(X, 1) = H_1(\text{the complex of Remark 0GU9}) \]

We use the supscript ${}^ M$ to distinguish our notation from the higher chow groups defined in the literature, e.g., in the papers by Spencer Bloch ([Bloch] and [Bloch-moving]). Let $U \subset X$ be open with complement $Y \subset X$ (viewed as reduced closed subscheme). Then we find a split short exact sequence

\[ 0 \to \bigoplus \nolimits _{y \in Y, \delta (y) = k + i}' K_ i^ M(\kappa (y)) \to \bigoplus \nolimits _{x \in X, \delta (x) = k + i}' K_ i^ M(\kappa (x)) \to \bigoplus \nolimits _{u \in U, \delta (u) = k + i}' K_ i^ M(\kappa (u)) \to 0 \]

for $i = 2, 1, 0$ compatible with the boundary maps in the complexes of Remark 42.27.2. Applying the snake lemma (see Homology, Lemma 12.13.6) we obtain a six term exact sequence

\[ \mathop{\mathrm{CH}}\nolimits ^ M_ k(Y, 1) \to \mathop{\mathrm{CH}}\nolimits ^ M_ k(X, 1) \to \mathop{\mathrm{CH}}\nolimits ^ M_ k(U, 1) \to \mathop{\mathrm{CH}}\nolimits _ k(Y) \to \mathop{\mathrm{CH}}\nolimits _ k(X) \to \mathop{\mathrm{CH}}\nolimits _ k(U) \to 0 \]

extending the canonical exact sequence of Lemma 42.19.3. With some work, one may also define flat pullback and proper pushforward for the first higher chow group $\mathop{\mathrm{CH}}\nolimits ^ M_ k(X, 1)$. We will return to this later (insert future reference here).

Comments (2)

Comment #2981 by Xia on

There seems to be some typos in the first displayed formula in the proof of Lemma 41.26.1 and the bracket after it.

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